We consider a given time of analysis and the population of exposed patients who will eventually experience the adverse drug reaction before they die. Let X be the time-to-onset of the adverse drug reaction of interest in that population and F its cumulative distribution function one is willing to estimate. Observations arising from n reported cases are (x ₁,t ₁),(x ₂,t ₂),…,(x _n ,t _n ), where x _i is the time-to-onset calculated as the lag between the time of the occurrence of the reaction and the time of initiation of treatment, and t _i is the truncation time calculated as the lag between the time of analysis and the time of initiation of treatment. Let t ^∗ be the maximum of the observed truncation times. All observed data meet the condition x _i ≤ t _i .

We consider a parametric model for the time-to-onset X, with cumulative distribution function F (x; θ) and density f(x; θ), and derive the following maximum likelihood estimations of θ.

When right truncation, i.e. the condition x _i ≤ t _i , is ignored, the likelihood of the sample is written as:

L 1 ( x 1 , x 2 , … , x n ; θ ) = ∏ i = 1 n f ( x i ; θ ) ;

maximizing this likelihood yields the naive estimator of θ.

When right truncation is considered, the likelihood is modified. Observed times-to-onset consist of n independent realizations of random variables with respective distribution the conditional distribution of X _i given {X _i ≤ t _i }, that is with cumulative distribution function F ( x i ; θ ) F ( t i ; θ ) and density f ( x i ; θ ) F ( t i ; θ ) . The likelihood is now written as:

L 2 ( x 1 , x 2 , … , x n , t 1 , t 2 , … , t n ; θ ) = ∏ i = 1 n f ( x i ; θ ) F ( t i ; θ ) ;

the maximum likelihood estimator from this likelihood, θ ̂ TBE , is the proper estimation of θ and is called the truncation-based estimator (TBE).

The non-parametric maximum likelihood estimation for right-truncated data was developed and used to estimate the incubation period distribution for AIDS 21 22 . However, in a non-parametric setting, one can only estimate the distribution function conditional on the time to event as being less than t ^∗ :

F ( x ) F ( t ∗ ) ̂ = ∏ v j > x 1 - n j N j ,

where the v _j ’s are the m distinct values of the x _i ’s, i =1,…,n, taken by n j = ∑ i = 1 n I ( X i = v j ) patients and N j = ∑ i = 1 n I ( X i ≤ v j ≤ t i ) for 1 ≤ j ≤ m, I denoting the indicator function. The unconditional distribution function is not identifiable, as F (t ^∗) is not known and cannot be estimated from the data.

In a parametric framework, the unconditional distribution is completely specified by a parameter θ of finite dimension. Maximum likelihood estimation of the parameter of interest can be conducted with the conditional distributions that describe the observations and the unconditional distribution can be estimated secondarily by F ( x ; θ ̂ TBE ) . Hence parametric maximum likelihood estimation is potentially more useful than non-parametric estimation since the unconditional distribution is of interest for pharmacovigilance purposes 18 20 .

Some adverse reactions have a very short time-to-onset, from several minutes to several hours after the beginning of treatment. Others occur only after several days, weeks, months or even years of exposure. This variation depends on numerous factors such as the pharmacokinetics of the drug and its metabolites, or the pathophysiological mechanism of the effect. The multiplicity of the underlying mechanisms results in a range of possible hazard functions that can be observed in pharmacovigilance 23 . The simplest model is given by a constant hazard function of time; the corresponding distribution is the exponential distribution with a rate parameter λ. Effects may also have an early or a late onset, the latter being the case for instance, when the rate of occurrence of the adverse reaction depends on the duration of exposure. Two distribution families among others make it possible to handle a wide range of hazard functions: the Weibull distributions and the log-logistic distributions (Table 1). Both are defined with two scalar parameters (λ,β); λ is the scale parameter and β is the shape parameter. The hazard function for the Weibull model is increasing if β > 1, decreasing if β < 1 and constant if β = 1 where it reduces to the exponential distribution. The hazard function for the log-logistic model is decreasing if β < 1 and has a single maximum if β > 1. We therefore consider the families of the exponential, Weibull and log-logistic distributions.

The times-to-onset were generated from these three distributions. Two values of λ were considered for the exponential distribution: 0.05 and 1. The same values were used for the scale parameter λ of the Weibull and log-logistic distributions. For the shape parameter β, the values 0.5 and 2 were chosen. The truncation times were uniformly distributed in [0,τ]. Survival and truncation times were independently generated. For a chosen value of p, with p representing the probability of X falling within the observable values interval [0, τ], the parameter τ was determined as P (X< τ) = p. The probability 1 - p is also a lower bound of the actual proportion of truncated data P (X> T), the truncation time T being randomly generated. The probability p was chosen in {0.25, 0.50, 0.80}. The sample size n was chosen in {100, 500}. For each drawn pair (X,T), if the time-to-onset was shorter than the truncation time, then the pair was included in the data. If not, another pair (X,T) was generated. Pairs were generated until the sample size of observations included was equal to n.

Parametric likelihood maximization with and without considering right truncation were performed for each generated sample. An iterative algorithm is necessary to solve this optimization problem except for the naive exponential estimation. Calculations were made with the R 24 function maxLik from the package maxLik. For each set of simulation parameters, 1000 replications were run.

We analyzed 64 French cases of lymphoma that occurred after anti TNF- α treatment using the national pharmacovigilance database at the date of February 1, 2010 25 . The population included patients suffering from rheumatoid arthritis, Crohn’s disease, ankylosing spondylitis, psoriatic arthritis, psoriasis, Sjögren’s syndrome, dermatomyositis, polymyositis or polyarthropathy and exposed to one or (successively) more of the three anti TNF- α available at the study date: etanercept, adalimumab and infliximab. The occurrence of a malignant lymphoma was confirmed by histopathological analysis. Marketing authorization was obtained in August 1999 for infliximab, in September 2002 for etanercept and in September 2003 for adalimumab. These 64 adverse effects occurred between July 2001 and October 2009. None of the survival or truncation times was missing in the database. The observed maximum truncation time was 529 weeks.

All anti TNF-agents taken together, we derived the parametric maximum likelihood estimates and secondarily corresponding estimated mean times, with and without considering right truncation, for the exponential, Weibull and log-logistic distributions. For completeness, we also derived the non-parametric maximum likelihood estimation.

The French pharmacovigilance database is developed by the French drug agency (Agence Nationale de Sécurité du Médicament et des produits de santé, ANSM) and is not publicly available. It is build up and used on an ongoing basis by the network of regional pharmacovigilance centres, which have a direct access to the data. This set of data has already been extracted for another study 25 with the authorization of the ANSM and the network of regional centres, according to the internal rule.

For each set of simulations parameters, for both approaches and for both parameters, the bias and the mean squared error of the parametric maximum likelihood estimator, based on the 1000 replications, were calculated as well as the proportion of replications where the estimate is larger than the true value. As the iterative algorithm may fail to find a maximum, those three quantities were actually calculated on the replications where there was no problem of maximization. The mean squared error is a measure of the dispersion of the estimator around the true value of the parameter - the smaller the better - and is used for global comparative purposes between two estimation procedures, as it incorporates both the variance of the estimator and its bias. The proportion of replications where the estimate is larger than the true value makes it possible to know if the estimators tend to overestimate or underestimate systematically the true value of the parameter.

Table 8 presents the estimates of the parameters for the three models and both approaches. There was no problem of maximization. The naive estimates are always larger than the truncation-based estimates. From the simulation results, it might be thought that the naive estimator overestimates the true values of parameters λ and β, and that the size of the bias is related to the unknown probability p. Estimations of the parameters for the truncation-based approach make it possible to estimate p by calculating F ( t ∗ = 529 ; θ ̂ TBE ) . However, estimates of p are different according to the model (Table 8). In particular, for the Weibull model, the estimate is large ( p ̂ = 0.98). The larger is p ̂ , the closer are the naive and the truncation-based estimates.

^*95% confidence intervals calculated using BCa simple bootstrap method based on 5000 replicates.

p ̂ = F ( t ∗ = 529 ; ( λ ̂ TBE , β ̂ TBE ) ) .

Abbreviations : TBE truncation-based estimator.

Figure 2 shows the non-parametric maximum likelihood estimation of the conditional survival function, F ( x ) F ( 529 ) ̂ , and the parametric maximum likelihood estimation of the conditional, F ( x ; θ ̂ TBE ) F ( 529 ; θ ̂ TBE ) , and unconditional, F ( x ; θ ̂ TBE ) , survival functions for the truncation-based approach for these data. The estimations of the conditional survival functions are always closer to the non-parametric estimation than the estimations of the unconditional survival functions. The conditional and unconditional estimations of the Weibull survival functions are almost similar because the estimate of p is about 1. This figure shows that the estimation of the conditional Weibull survival function is closer to the non-parametric maximum likelihood estimation of the conditional survival function than the estimations of the conditional exponential and conditional log-logistic survival functions. Thus, Weibull could be a reasonable candidate model to describe the data.

Figure 3 shows the parametric maximum likelihood estimation of the unconditional survival function for both approaches. The distance between both survivals, naive and truncation-based, decreases with the estimated probability p ̂ (in the order: exponential, log-logistic and Weibull). Furthermore, the survival functions from the truncation-based estimates are always above the survival functions from the naive estimates, which is consistent with the naive estimator overestimating the true values of the parameters λ and β. Even for the Weibull model, i.e. the model with the largest p ̂ , the estimated expected time-to-onset would be 135 weeks with the naive approach and 193 weeks with the truncation-based estimates, which corresponds to a markedly large gap (Table 8). For completeness, we also calculated the 95% simple bootstrap confidence intervals of the expected time (BCa method) 26 27 based on 5000 bootstrap samples, for the truncation-based approach. They do not include the naive estimated mean time, whatever the fitted model, and even though these confidence intervals are extremely wide.